In piezoelectric resonators in which piezoelectric resonator elements each including a pair of electrode films with a piezoelectric film are supported on substrates so as to be acoustically isolated from the substrates and operating in a thickness-extensional vibration mode, a transverse vibration mode generates waves that propagate in a planar direction when piezoelectric resonator elements vibrate in the thickness direction, causing the generation of a spurious wave.
To suppress such a spurious wave, for example, as shown in FIGS. 1 and 2, a piezoelectric resonator is reported to have a free-form vibrating region and a bump along the periphery of the vibrating region. That is, a bottom electrode 110 and a piezoelectric layer 100 are each disposed on a SiO2 support layer 200 serving as a substrate and each extend across the full width of the resonator. Boundary regions 2 and 4 and a central portion 3 of a resonator structure 1000 are covered with a top electrode 120. The boundary regions 2 and 4 of the resonator structure 1000 are formed by increasing the thickness of the top electrode layer 120 on the periphery of the electrically excitable vibrating region. The central portion 3 of a resonator 1200 has a free form. Arrows in FIG. 2 indicate scale. The scale in the horizontal direction differs from that in the vertical direction.
In the case where the bump is formed between the free-form central portion 3 and the boundary regions 2 and 4, a long-wavelength spurious Lamb wave and a short-wavelength spurious Lamb wave are simultaneously suppressed. That is, the bump is formed in such a manner that the thickness of the periphery of the vibrating region is larger than the thickness of the central portion, thereby suppressing the long-wavelength wave. The planar shape of the vibrating region is a free form. This suppresses the short-wavelength wave.
The reason for the edge (boundary regions) of the vibrating region having a thickness larger than that of the central region is that the ZnO exhibiting the dispersion property of type I is used. The type I is one of general forms of the dispersion relation k (ω) depicted in FIG. 3. In FIG. 3, the vertical axis represents the angular frequency ω. The horizontal axis on the right side of the vertical axis represents real values of wave number k. On the left side of the vertical axis, the wave number is imaginary. In type I dispersion, the wave number k is real when the angular frequency ω is above the cut-off angular frequency ωc of the plate. In type II dispersion, the wave number k is real when the angular frequency ω is below the cut-off angular frequency ωc of the plate (for example, Patent Document 1).    Patent Document 1: PCT Japanese Translation Patent Publication No. 2003-505906
In FIG. 3, only a small-wave-number (long-wavelength) region is calculated. In a structure including a piezoelectric film having the type II dispersion property depicted in FIG. 3, for example, a dispersion curve determined by more detailed calculation is shown in FIG. 4. In FIG. 4, the vertical axis represents the frequency. The horizontal axis represents b/λ obtained by normalizing the thickness b of the piezoelectric film by the wavelength λ of the wave propagating in a planar direction. That is, FIG. 4 shows the dispersion curve of the real wave number part. Reference numeral 6a represents the case of a wavelength that is about five times the thickness of the piezoelectric film. Reference numeral 6b represents the case of a wavelength that is about twice the thickness of the piezoelectric film. Reference numeral 8 represents a resonant frequency. The graph demonstrates that long-wavelength Lamb waves 7a and short-wavelength Lamb waves 7b are both present at frequencies near the resonant frequency.
A reduction in the thickness of the cross section of a frame-like zone at the edge of the vibrating region eliminates the generation of a long-wavelength spurious mode in a region where the wavelength represented by reference numeral 7a is about five or more times the thickness of the piezoelectric film in FIG. 4. In the case of the dispersion curve as shown in FIG. 4, however, in the vicinity of the resonant frequency 8, a short-wavelength spurious mode is present in a region where the wavelength represented by reference numeral 7b is equal to or less than about twice the thickness of the piezoelectric film in FIG. 4 in addition to the long-wavelength spurious mode.
The short-wavelength spurious mode can be suppressed by forming the planar shape of the resonator into a free form as shown in FIG. 1. However, the resonator having the shape is difficult to arrange and is not suitable for the miniaturization of the piezoelectric resonator. The planar shape of the resonator also affects the layout of a filter and the size of a chip; hence, a substantially rectangular shape is preferred.
The origin of the short-wavelength spurious radiation is insufficiently analyzed. Thus, an optimum planar shape of the vibrating region is not found.